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Cambridge University Press eBooks

Floquet theory

C. J. Joachain, Niels J. Kylstra, R. M. Potvliege

Published Dec 15, 2011• 0 citations
Publisher

Abstract

In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively. We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies , which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.

DOI
10.1017/cbo9780511993459.005

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In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively. We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies , which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.
The Floquet theorem allows one to reformulate periodic time-dependent problems such as the interaction of a many-body system with a laser field in terms of time-independent, field-dressed states, also known as Floquet states. If this was possible for density-functional theory as well, one could reduce in such cases time-dependent density-functional theory to a time-independent Floquet density-functional theory. We analyze under which conditions the Floquet theorem is applicable in a density-functional framework. By employing numerical ab initio solutions of the interacting time-dependent Schr\"odinger equation with time-periodic external potentials we show that the exact effective potential in the corresponding Kohn-Sham equation is not unconditionally periodic. Whenever several Floquet states in the interacting system are involved in a physical process the corresponding Hartree-exchange-correlation potential is not periodic with the external frequency only. Using an analytically solvable example we demonstrate that, in general, the periodicity of the time-dependent Kohn-Sham Hamiltonian cannot be restored by choosing a different initial state. Only if the external periodic potential is sufficiently weak such that the initial state of the interacting system evolves adiabatically to a single, field-dressed state, the resulting Kohn-Sham system admits the application of the Floquet theorem.
Many-mode Floquet theory [T.-S. Ho, S.-I. Chu, and J. V. Tietz, Chem. Phys. Lett. \n96, 464 (1983)] was designed as an extension of Floquet theory suitable for solving the \ntime-dependent Schrodinger equation with multiple periodicities, however its limitations \nare not well understood. I show that for two commensurate frequencies (integer multiples of \na common frequency), many-mode Floquet theory always produces an exact expression for \nthe time evolution of a system, despite only part of the eigenvalue spectrum being directly \nrelevant. I show that the rest of the spectrum corresponds to eigenvalues of the same \nsystem but at other values of the relative phase between the bichromatic field components. \n \n \nI show by using a Floquet perturbative analysis that dressing a Rydberg atom with a \nbichromatic field with frequency components ω2 and ω1, such that ω2 = 2ω1, can induce a \npermanent dipole moment (first order energy shift with dc electric field) without a dc bias \nfield. With frequency ω1 = 2π5.997GHz, ω2 = 2ω1 and field strengths of Eac1 = 0.1 V/cm \nand Eac2 = 0.05 V/cm, a permanent dipole moment of magnitude 44.06 MHz/(V/cm) is \ninduced in the dressed 65s1/2 state of ⁸⁵Rb. The permanent dipole moment depends on the \nrelative phase between the fields and can be made to be zero at certain values of phase.
The dressed atom approach provides a tool to investigate the dynamics of an atom-laser system by fully retaining the quantum nature of the coherent mode. In its standard derivation, the internal atom-laser evolution is described within the rotating-wave approximation, which determines a doublet structure of the spectrum and the peculiar fluorescence triplet in the steady state. However, the rotating wave approximation may fail to apply to atomic systems subject to femtosecond light pulses, light-matter systems in the strong-coupling regime or sustaining permanent dipole moments. This work aims to demonstrate how the general features of the steady-state radiative cascade are affected by the interaction of the dressed atom with propagating radiation modes. Rather than focusing on a specific model, we analyze how these features depend on the parameters characterizing the dressed eigenstates in arbitrary atom-laser dynamics, given that a set of general hypotheses is satisfied. Our findings clarify the general conditions in which a description of the radiative cascade in terms of transition between dressed states is self-consistent. We provide a guideline to determine the properties of photon emission for any atom-laser interaction model, which can be particularly relevant when the model should be tailored to enhance a specific line. We apply the general results to the case in which a permanent dipole moment is a source of low-energy emission, whose frequency is of the order of the Rabi coupling.
A theory of the absorption of a laser field by an atomic or condensed matter medium is presented for the case where the medium is also interacting with a strong electromagnetic field. The rotating wave approximation is not assumed for the latter. It is shown that in the weak probe limit the Lindblad master equation reduces to a smaller system of linear equations for the relevant steady-state coherences. In this limit, the complex susceptibility of the medium can be expressed in terms of individual contributions of decaying dressed states, the latter being eigenstates of a non-Hermitian Floquet Hamiltonian.
Energy Approach to Atoms in a Laser Field and Quantum Dynamics with Laser Pulses of Different Shapes 161 strong nonlinear coupling of radiation with atoms which can not fully explained by the modern theoretical models.In any case the problem requires a consistent quantum electrodynamic approach.Another important topic is a problem of governing and control of non-linear processes in a stochastic, multi-mode laser field (Grance, 1981;Lompre et al, 1981;Zoller, 1982; Glushkov & Ivanov, 1992).The principal aim of quantum coherent control is to steer a quantum system towards a desired final state through interaction with light while simultaneously inhibiting paths leading to undesirable outcomes.This type of quantum interference is inherent in non-linear multiphoton processes.Controlling mechanisms have been proposed and demonstrated for atomic, molecular and solid-state systems (Goldansky-Letokhov, 1974, Letokhov, 1977;Delone-Kraynov, 1984).Theoretical studies of the laser-atom nonlinear interactions are often based on solving the time-dependent Schrödinger equation or using the time-independent Floquet formalism or special perturbation theories (Brändas &
In Chap. 1 we introduced the Schrodinger equation and gave solutions for a number of simple cases. The general solution was then obtained by a linear superposition of these solutions. In our work, we are particularly concerned with the way in which atomic systems interact with electromagnetic radiation. These systems are still described by wave functions given by superpositions of simple solutions, but the expansion coefficients Cn (probability amplitudes) become functions of time. In this chapter we derive equations of motion for the expansion coefficients using Schrddinger’s equation and introduce the electric-dipole interaction energy. Then, specializing to a two-level system and a monochromatic electromagnetic field, we solve the equations of motion both by time-dependent perturbation theory and by the more exact Rabi method.